Ork — ExteAisions of Fourier'' s and the Bessel-Fourier Theorems. 15 



= ^ (;/, (tc - s) + i// (,-r + e)), a < X <b; (5) 



or = \p (a), it X = a ; 

 or = ip (b), if « = &. 



I shall find it convenient to refer to the left-hand member of (5) 

 indifferently as an integral or as the equivalent sum of residues. 



The equation of type (2) derivable from this is valid at the boundaries 

 a, b ; it is valid also at discontinuities in i//, provided xp' is continuous ; but at 

 internal discontinuities in \p' the value of the left-hand member of (2) is not 

 simply X, but 



The truth of (5) should be obvious.f In establishing the equation of 

 type (2) the only point of diificulty lies in showing that the part of the 

 left-hand member which involves xjj is zero. Perhaps the simplest manner of 

 ■surmounting this, except for the boundaries, is to note that, at any time t for 

 which ,x ± d both lie between a, b, the value of the left-hand member of (5) is 



^\\p{x + d) + \p (x - (i)}. 



The single integrals in (5) which involve the boundary values affect the 

 initial values neither of (p nor of d(j>/dt at points in the interior of the range ; 

 if we omit them and make X" ^> ^^ obtain the expansion of the former 

 paper for \p; if we omit them and make t// = in the corresponding equation 

 of type (2), we obtain the expansion of the former paper for x- 



Akt. 4. The Fxpansio^is still Not Unique : Terms which may be added. 



These expansions are, however, not unique. We may, in fact, add to the 

 left-hand member of (5) 



^^) - 0^(i-)F,{-f,)F{^ty0^i''-'>)F(f,)F{-^) 



(6) 

 where Fifi), /(fi) are polynomials of orders not exceeding p - 3, q ~ 3 

 respectively, without affecting the validity of (1), (2), or altering their type. 



* If an elastic string is released from rest with a transverse displacement \f/, any point a; at -which 

 ij/' is discontinuous acquires immediately a velocity 



and it seems impossible to give any analytical expression for the displacement at time t which will 

 not indicate this velocity as existing initially. 



t In a great deal of what follows it seems undesirable to give the reasoning fully, as to do so 

 would add considerably to the length of the Paper, unduly great as it is, 



